Stephen Paul King (stephenk1@home.com)
Fri, 26 Mar 1999 16:00:57 GMT
On Fri, 26 Mar 1999 01:34:26 GMT, Minhyong Kim <kim@math.arizona.edu>
wrote:
>Hi John,
>
>Just one comment on Poincare duality at the level of
>forms (as you can see, I still haven't completely figured
>out my news-reader, so I'm sending this as a separate
>post rather than continuing the thread):
>It doesn't seem correct that one needs the
>Hodge star operator to get the duality. The duality
>comes from the fact that p-forms and n-p forms
>pair to give an n-form which can then be integrated
>over the oriented manifold. One doesn't need a metric
>anywhere to get this duality. The metric and the
>star operator gives us an *isomorphism* between
>H^p and H^{n-p}. This distinction should be made
>because it's a case of a rather important principle
>that a duality can somtimes be turned into an isomorphism
>when given extra structure. This sort of stuff is
>your speciality, so I couldn't resist the temptation
>to catch you off-guard!
>
>Minhyong
>
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